5060

-3*(x+y+z)*(x^2+x*y-x^2*y+y^2-x*y^2+x*z-x^2*z+y*z-y^2*z+z^2-x*z^2-y*z^2)^2+(x^2+x*y+y^2+x*z+y*z+z^2)^2*(-(x*z*(y^2-x*z))+3*(x*(-1+y)*(-1+z)+(-1+x)*y*(-1+z)+(-1+x)*(-1+y)*z)-y*z*(x^2-y*z)-x*y*(-(x*y)+z^2))//Factor

在１楼题设下，对参量k>=1，有：k^2*s[b*c*(a^2-b*c)]≤(k+2)*s[a*(b-1)*(c-1)]+(2*k+1)*s[b*c*(a-1)^2]。

使用无敌的pqr法
In[371]:= -With[{a = x, b = y, c = z},
k^2*
s[b*c*(a^2 - b*c)] - ((k + 2)*s[a*(b - 1)*(c - 1)] + (2*k + 1)*
s[b*c*(a - 1)^2])
] /. k -> 1 + k // pqr // Collect[#, r, Factor] &
Out[371]=
3 p + k p - 3 q + q^2 + 2 k q^2 +
k^2 q^2 + (-9 - 9 k - 4 k p - 3 k^2 p) r
关于 r 线性且系数负的. 设 {q -> (p^2 - t^2)/3},t>0, 则 1/27 (p - 2 t) (p + t)^2 <= r <= 1/27 (p - t)^2 (p + 2 t)
只用考虑 r = 1/27 (p - t)^2 (p + 2 t) ,
In[380]:=
3 p + k p - 3 q + q^2 + 2 k q^2 +
k^2 q^2 + (-9 - 9 k - 4 k p - 3 k^2 p) r /. {q -> (p^2 - t^2)/
3} /. r -> 1/27 (p - t)^2 (p + 2 t) // Factor
Out[380]= 1/27 (81 p + 27 k p - 27 p^2 - 9 p^3 - 9 k p^3 + 3 p^4 +
2 k p^4 + 27 t^2 + 27 p t^2 + 27 k p t^2 - 6 p^2 t^2 +
3 k^2 p^2 t^2 - 18 t^3 - 18 k t^3 - 8 k p t^3 - 6 k^2 p t^3 +
3 t^4 + 6 k t^4 + 3 k^2 t^4)
验证
In[383]:= FindInstance[{81 p + 27 k p - 27 p^2 - 9 p^3 - 9 k p^3 +
3 p^4 + 2 k p^4 + 27 t^2 + 27 p t^2 + 27 k p t^2 - 6 p^2 t^2 +
3 k^2 p^2 t^2 - 18 t^3 - 18 k t^3 - 8 k p t^3 - 6 k^2 p t^3 +
3 t^4 + 6 k t^4 + 3 k^2 t^4 < 0, {k, p, t} > 0}, {k, p, q, t}]
Out[383]= {}
确认没问题. 配方
(81 p+27 k p-27 p^2-9 p^3-9 k p^3+3 p^4+2 k p^4+27 t^2+27 p t^2+27 k p t^2-6 p^2 t^2+3 k^2 p^2 t^2-18 t^3-18 k t^3-8 k p t^3-6 k^2 p t^3+3 t^4+6 k t^4+3 k^2 t^4)(p+1)==10/3 p (-3+p-t)^2+5/2 k p (-3+p-t)^2+11/3 p^2 (-3+p-t)^2+3 k p^2 (-3+p-t)^2+4 k p (p-t)^2 t^2+1/3 p (-12-5 p+3 p^2+5 t-3 t^2)^2+1/2 k p (-3-5 p+2 p^2+5 t-2 t^2)^2+1/3 p (3-p+t+3 k p t-3 k t^2)^2+1/3 (3 p-p^2+9 t+4 p t+3 k p t-3 t^2-3 k t^2)^2//Factor

看看哪出问题了
In[468]:= Module[{a , b, c, n, vars},
n = 4;
vars = Subscript[x, #] & /@ Range[n];
{a, b, c} = SymmetricPolynomial[#, vars] & /@ Range[3] // Echo;
FindInstance[{3 (a + 3 c - 2 b) + b^2 - 3 a c < 0, Total[vars] >= 0},
vars]
]
{Subscript[x, 1]+Subscript[x, 2]+Subscript[x, 3]+Subscript[x, 4],Subscript[x, 1] Subscript[x, 2]+Subscript[x, 1] Subscript[x, 3]+Subscript[x, 2] Subscript[x, 3]+Subscript[x, 1] Subscript[x, 4]+Subscript[x, 2] Subscript[x, 4]+Subscript[x, 3] Subscript[x, 4],Subscript[x, 1] Subscript[x, 2] Subscript[x, 3]+Subscript[x, 1] Subscript[x, 2] Subscript[x, 4]+Subscript[x, 1] Subscript[x, 3] Subscript[x, 4]+Subscript[x, 2] Subscript[x, 3] Subscript[x, 4]}
Out[468]= {{Subscript[x, 1] -> 1, Subscript[x, 2] -> 3/2,
Subscript[x, 3] -> 7/5, Subscript[x, 4] -> 6/5}}

６楼修正：
对实数a、b、c，当a＋b＋c⩾0时，有9*Σ[a*((1-b)*(1-c))]+Σ[(b^2+b*c+c^2)*((a-b)*(a-c))]≥0；
对实数a、b、c、d，当a＋b＋c＋d⩾0时，有24*Σ[a*((1-b)*(1-c)+(1-b)*(1-d)+(1-c)*(1-d))]+Σ[(b^2+c^2+d^2+b*c+b*d+c*d)*((a-b)*(a-c)+(a-b)*(a-d)+(a-c)*(a-d))]≥0；
对实数a、b、c、d、e，当a＋b＋c＋d＋e⩾0时，猜有50*Σ[a*((1-b)*(1-c)+(1-b)*(1-d)+(1-c)*(1-d)+(1-b)*(1-e)+(1-c)*(1-e)+(1-d)*(1-e))]+Σ[(b^2+c^2+d^2+e^2+b*c+b*d+c*d+b*e+c*e+d*e)*((a-b)*(a-c)+(a-b)*(a-d)+(a-c)*(a-d)+(a-b)*(a-e)+(a-c)*(a-e)+(a-d)*(a-e))]≥0；
对实数a、b、c、d、e、f，当a＋b＋c＋d＋e＋f⩾0时，猜有90*Σ[a*((1-b)*(1-c)+(1-b)*(1-d)+(1-c)*(1-d)+(1-b)*(1-e)+(1-c)*(1-e)+(1-d)*(1-e)+(1-b)*(1-f)+(1-c)*(1-f)+(1-d)*(1-f)+(1-e)*(1-f))]+Σ[(b^2+c^2+d^2+e^2+f^2+b*c+b*d+c*d+b*e+c*e+d*e+b*f+c*f+d*f+e*f)*((a-b)*(a-c)+(a-b)*(a-d)+(a-c)*(a-d)+(a-b)*(a-e)+(a-c)*(a-e)+(a-d)*(a-e)+(a-b)*(a-f)+(a-c)*(a-f)+(a-d)*(a-f)+(a-e)*(a-f))]≥0；
对实数a、b、c、d、e、f、g，当a＋b＋c＋d＋e＋f＋g⩾0时，猜有147*Σ[a*((1-b)*(1-c)+(1-b)*(1-d)+(1-c)*(1-d)+(1-b)*(1-e)+(1-c)*(1-e)+(1-d)*(1-e)+(1-b)*(1-f)+(1-c)*(1-f)+(1-d)*(1-f)+(1-e)*(1-f)+(1-b)*(1-g)+(1-c)*(1-g)+(1-d)*(1-g)+(1-e)*(1-g)+(1-f)*(1-g))]+Σ[(b^2+c^2+d^2+e^2+f^2+g^2+b*c+b*d+c*d+b*e+c*e+d*e+b*f+c*f+d*f+e*f+b*g+c*g+d*g+e*g+f*g)*((a-b)*(a-c)+(a-b)*(a-d)+(a-c)*(a-d)+(a-b)*(a-e)+(a-c)*(a-e)+(a-d)*(a-e)+(a-b)*(a-f)+(a-c)*(a-f)+(a-d)*(a-f)+(a-e)*(a-f)+(a-b)*(a-g)+(a-c)*(a-g)+(a-d)*(a-g)+(a-e)*(a-g)+(a-f)*(a-g))]≥0；
对实数a、b、c、d、e、f、g、h，当a＋b＋c＋d＋e＋f＋g＋h⩾0时，猜有224*Σ[a*((1-b)*(1-c)+(1-b)*(1-d)+(1-c)*(1-d)+(1-b)*(1-e)+(1-c)*(1-e)+(1-d)*(1-e)+(1-b)*(1-f)+(1-c)*(1-f)+(1-d)*(1-f)+(1-e)*(1-f)+(1-b)*(1-g)+(1-c)*(1-g)+(1-d)*(1-g)+(1-e)*(1-g)+(1-f)*(1-g)+(1-b)*(1-h)+(1-c)*(1-h)+(1-d)*(1-h)+(1-e)*(1-h)+(1-f)*(1-h)+(1-g)*(1-h))]+Σ[(b^2+c^2+d^2+e^2+f^2+g^2+h^2+b*c+b*d+c*d+b*e+c*e+d*e+b*f+c*f+d*f+e*f+b*g+c*g+d*g+e*g+f*g+b*h+c*h+d*h+e*h+f*h+g*h)*((a-b)*(a-c)+(a-b)*(a-d)+(a-c)*(a-d)+(a-b)*(a-e)+(a-c)*(a-e)+(a-d)*(a-e)+(a-b)*(a-f)+(a-c)*(a-f)+(a-d)*(a-f)+(a-e)*(a-f)+(a-b)*(a-g)+(a-c)*(a-g)+(a-d)*(a-g)+(a-e)*(a-g)+(a-f)*(a-g)+(a-b)*(a-h)+(a-c)*(a-h)+(a-d)*(a-h)+(a-e)*(a-h)+(a-f)*(a-h)+(a-g)*(a-h))]≥0；
一般地，

8楼：
令 f(x) = (x - x1)*(x - x2)*...(x - xn) = x^n -s1*x^(n-1) + s2*x^(n-2) - s3*x^(n-3) + ...
则由罗尔定理 f的(n-3)阶导有3个实根。
即： n!/6 * x³ - (n-1)!/2 * s1 * x² + (n-2)! * s2 * x - (n-3)! * s3 有三个实根，设为a,b,c
则由韦达定理， s1 = n/3*(a+b+c), s2=n(n-1)/6*(ab+bc+ca), s3=n(n-1)(n-2)/6*abc
代回去，发现跟三元长得一样，证毕

2L 的还是安卓了点，下面的配法更苹果
In[134]:=
g = s[ (x - y)^2 (-1 + z)^2];
g = (s[(x - y)^2 (z - 1)]^2 + 3 (x - y)^2 (x - z)^2 (y - z)^2)/
s[(x - y)^2];
3/4 (x - y)^2 (x - z)^2 (y - z)^2 +
1/4 (-6 x - 6 y + 6 x y + x^2 y + x y^2 - 6 z + 6 x z + x^2 z +
6 y z - 6 x y z + y^2 z + x z^2 + y z^2)^2 +
9 (-x - y + x y - z + x z + y z)^2 + 3/2 s[x] g // Factor
Out[136]= (6 x + x^2 + 6 y - x y + y^2 + 6 z - x z - y z +
z^2) (3 x + 3 y - 6 x y + x^2 y^2 + 3 z - 6 x z - 6 y z + 9 x y z -
x^2 y z - x y^2 z + x^2 z^2 - x y z^2 + y^2 z^2)

８楼亖元时的加强：
对实数a、b、c、d，若a＋b＋c＋d⩾0，则有24*Σ[a*((1-b)*(1-c)+(1-b)*(1-d)+(1-c)*(1-d))]+6*((a*d-b*c)^2+(a*c-b*d)^2+(a*b-c*d)^2)+1/4*((a-d)^2*(b-c)^2+(a-c)^2*(b-d)^2+(a-b)^2*(c-d)^2)>=0。

11 L ，1=v算关于v的判别式，证明 F = 13689 a^6 b^4-25870 a^5 b^5+13689 a^4 b^6-2106 a^6 b^3 c+196 a^5 b^4 c+196 a^4 b^5 c-2106 a^3 b^6 c+27459 a^6 b^2 c^2-27526 a^5 b^3 c^2+38503 a^4 b^4 c^2-27526 a^3 b^5 c^2+27459 a^2 b^6 c^2-2106 a^6 b c^3-27526 a^5 b^2 c^3+16026 a^4 b^3 c^3+16026 a^3 b^4 c^3-27526 a^2 b^5 c^3-2106 a b^6 c^3+13689 a^6 c^4+196 a^5 b c^4+38503 a^4 b^2 c^4+16026 a^3 b^3 c^4+38503 a^2 b^4 c^4+196 a b^5 c^4+13689 b^6 c^4-25870 a^5 c^5+196 a^4 b c^5-27526 a^3 b^2 c^5-27526 a^2 b^3 c^5+196 a b^4 c^5-25870 b^5 c^5+13689 a^4 c^6-2106 a^3 b c^6+27459 a^2 b^2 c^6-2106 a b^3 c^6+13689 b^4 c^6-2106 a^6 b^3 d+196 a^5 b^4 d+196 a^4 b^5 d-2106 a^3 b^6 d-1944 a^6 b^2 c d-58184 a^5 b^3 c d+65682 a^4 b^4 c d-58184 a^3 b^5 c d-1944 a^2 b^6 c d-1944 a^6 b c^2 d+1338 a^5 b^2 c^2 d+316 a^4 b^3 c^2 d+316 a^3 b^4 c^2 d+1338 a^2 b^5 c^2 d-1944 a b^6 c^2 d-2106 a^6 c^3 d-58184 a^5 b c^3 d+316 a^4 b^2 c^3 d-118440 a^3 b^3 c^3 d+316 a^2 b^4 c^3 d-58184 a b^5 c^3 d-2106 b^6 c^3 d+196 a^5 c^4 d+65682 a^4 b c^4 d+316 a^3 b^2 c^4 d+316 a^2 b^3 c^4 d+65682 a b^4 c^4 d+196 b^5 c^4 d+196 a^4 c^5 d-58184 a^3 b c^5 d+1338 a^2 b^2 c^5 d-58184 a b^3 c^5 d+196 b^4 c^5 d-2106 a^3 c^6 d-1944 a^2 b c^6 d-1944 a b^2 c^6 d-2106 b^3 c^6 d+27459 a^6 b^2 d^2-27526 a^5 b^3 d^2+38503 a^4 b^4 d^2-27526 a^3 b^5 d^2+27459 a^2 b^6 d^2-1944 a^6 b c d^2+1338 a^5 b^2 c d^2+316 a^4 b^3 c d^2+316 a^3 b^4 c d^2+1338 a^2 b^5 c d^2-1944 a b^6 c d^2+27459 a^6 c^2 d^2+1338 a^5 b c^2 d^2+152343 a^4 b^2 c^2 d^2-19708 a^3 b^3 c^2 d^2+152343 a^2 b^4 c^2 d^2+1338 a b^5 c^2 d^2+27459 b^6 c^2 d^2-27526 a^5 c^3 d^2+316 a^4 b c^3 d^2-19708 a^3 b^2 c^3 d^2-19708 a^2 b^3 c^3 d^2+316 a b^4 c^3 d^2-27526 b^5 c^3 d^2+38503 a^4 c^4 d^2+316 a^3 b c^4 d^2+152343 a^2 b^2 c^4 d^2+316 a b^3 c^4 d^2+38503 b^4 c^4 d^2-27526 a^3 c^5 d^2+1338 a^2 b c^5 d^2+1338 a b^2 c^5 d^2-27526 b^3 c^5 d^2+27459 a^2 c^6 d^2-1944 a b c^6 d^2+27459 b^2 c^6 d^2-2106 a^6 b d^3-27526 a^5 b^2 d^3+16026 a^4 b^3 d^3+16026 a^3 b^4 d^3-27526 a^2 b^5 d^3-2106 a b^6 d^3-2106 a^6 c d^3-58184 a^5 b c d^3+316 a^4 b^2 c d^3-118440 a^3 b^3 c d^3+316 a^2 b^4 c d^3-58184 a b^5 c d^3-2106 b^6 c d^3-27526 a^5 c^2 d^3+316 a^4 b c^2 d^3-19708 a^3 b^2 c^2 d^3-19708 a^2 b^3 c^2 d^3+316 a b^4 c^2 d^3-27526 b^5 c^2 d^3+16026 a^4 c^3 d^3-118440 a^3 b c^3 d^3-19708 a^2 b^2 c^3 d^3-118440 a b^3 c^3 d^3+16026 b^4 c^3 d^3+16026 a^3 c^4 d^3+316 a^2 b c^4 d^3+316 a b^2 c^4 d^3+16026 b^3 c^4 d^3-27526 a^2 c^5 d^3-58184 a b c^5 d^3-27526 b^2 c^5 d^3-2106 a c^6 d^3-2106 b c^6 d^3+13689 a^6 d^4+196 a^5 b d^4+38503 a^4 b^2 d^4+16026 a^3 b^3 d^4+38503 a^2 b^4 d^4+196 a b^5 d^4+13689 b^6 d^4+196 a^5 c d^4+65682 a^4 b c d^4+316 a^3 b^2 c d^4+316 a^2 b^3 c d^4+65682 a b^4 c d^4+196 b^5 c d^4+38503 a^4 c^2 d^4+316 a^3 b c^2 d^4+152343 a^2 b^2 c^2 d^4+316 a b^3 c^2 d^4+38503 b^4 c^2 d^4+16026 a^3 c^3 d^4+316 a^2 b c^3 d^4+316 a b^2 c^3 d^4+16026 b^3 c^3 d^4+38503 a^2 c^4 d^4+65682 a b c^4 d^4+38503 b^2 c^4 d^4+196 a c^5 d^4+196 b c^5 d^4+13689 c^6 d^4-25870 a^5 d^5+196 a^4 b d^5-27526 a^3 b^2 d^5-27526 a^2 b^3 d^5+196 a b^4 d^5-25870 b^5 d^5+196 a^4 c d^5-58184 a^3 b c d^5+1338 a^2 b^2 c d^5-58184 a b^3 c d^5+196 b^4 c d^5-27526 a^3 c^2 d^5+1338 a^2 b c^2 d^5+1338 a b^2 c^2 d^5-27526 b^3 c^2 d^5-27526 a^2 c^3 d^5-58184 a b c^3 d^5-27526 b^2 c^3 d^5+196 a c^4 d^5+196 b c^4 d^5-25870 c^5 d^5+13689 a^4 d^6-2106 a^3 b d^6+27459 a^2 b^2 d^6-2106 a b^3 d^6+13689 b^4 d^6-2106 a^3 c d^6-1944 a^2 b c d^6-1944 a b^2 c d^6-2106 b^3 c d^6+27459 a^2 c^2 d^6-1944 a b c^2 d^6+27459 b^2 c^2 d^6-2106 a c^3 d^6-2106 b c^3 d^6+13689 c^4 d^6>=0.

2L思路补遗：

请问楼主
l神跟豪神图片解答是什么软件工具啊🤔